fluid dynamic analysis of non-traditional supersonic projectiles

Fluid Dynamic Analysis of Non-TraditionalSupersonic Projectiles

Stuart McIlwain * Mahmood Khalid *

* Institute for Aerospace ResearchNational Research Council of Canada

Ottawa, ON K1A 0R6, Canada.E-mail: [email protected]

Received 11 July 2005.

NOMENCLATURE

A missile area (mm2)

CP surface pressure coefficient

CX axial force coefficient

CY normal force coefficient

CM pitching moment coefficient

D missile diameter (mm)

L missile length (mm)

Ma freestream Mach number

Re Reynolds number

S width of the sides of the square cross-section missile(mm)

x axial coordinate

y normal coordinate

y+ normalized distance from the surface

α angle of attack (°)

β yaw angle (°)

θ position on the missile surface (°)

φ roll angle (°)

INTRODUCTION

Over the last four decades, researchers have attempted toenhance the performance attributes of flight weapons, such

as low observablity, increased stability, and improvedaerodynamic properties, by considering projectiles with non-traditional cross sections and multiple sets of control surfaces.Objects with non-traditional-shaped cross sections also providethe potential for improved packaging of internal components.The cross sections often contain sharp corners, which give riseto vortex systems that may improve the lift characteriztics forcertain projectile orientations but may also interfere with thecontrol surfaces. A summary of prior studies of missile withnon-traditional shaped cross sections can be found in Birch andCleminson (2004). Multiple sets of control surfaces aim toimprove the controlability of the projectile. The additionalcontrol surfaces intensify complex vortex interactions withinthe flow field, which may enhance the performance of theprojectile, but may also induce catastrophic interference effectswith the downstream control surfaces.

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AbstractComputational fluid dynamics was used to predict thesupersonic flow fields around three different missileconfigurations: one with an elliptical cross section, onewith two sets of control fins, and one with a square crosssection. The angles of attack ranged from 5° to 24° and thefreestream Mach numbers ranged from 1.75 to 2.5. Thenumerical results were in good agreement with wind tunneldata for each missile, and the simulations captured theevolving vortex systems surrounding the three differentmissile configurations. Certain roll angles and angles ofattack produced vortex systems that interfered with thecontrol surfaces of the missile, which could lead tocatastrophic effects on the missile aerodynamics.

RésuméOn a utilisé la dynamique numérique des fluides pourprévoir les champs d’écoulement supersonique autour dedifférentes configurations de missile : une avec une coupetransversale elliptique, une avec deux ensembles d’ailettesde commande et une avec une coupe transversale carrée.Les angles d’attaque variaient de 5° à 24° et les nombres deMach en écoulement libre de l’air, de 1,75 à 2,5. Pourchaque missile, les résultats numériques obtenuscorrespondaient assez bien aux données obtenues ensoufflerie, et les simulations ont pu reproduire les systèmesde tourbillons en évolution autour des trois différentesconfigurations de missile. Certains angles d’inclinaison etd’attaque ont généré des systèmes de tourbillons qui ontperturbé les gouvernes du missile, ce qui aurait pu avoir deseffets catastrophiques sur les caractéristiquesaérodynamiques du missile.

Two recent studies conducted under the auspices of TheTechnical Cooperation Programme (TTCP) and the NATOResearch Technology Organisation (RTO) focused on theaerodynamic performance of realistic supersonic non-traditional missile configurations. The flow fields around non-traditional projectiles present challenges to computational fluiddynamics (CFD) algorithms due to the presence of complexvortex systems and vortex and (or) shock interactions.Therefore, both studies aimed to determine whether modernCFD solvers could resolve the complex vortex activity presentin the projectile flow fields by generating comprehensiveexperimental data bases that were used for code-validationpurposes. The aerodynamic performance of several missileswas examined for different flight conditions to determine theextent of the improvements offered by using non-traditionalcross sections and multiple sets of control surfaces. A summaryof the results from the RTO and KTa 2-19 studies can be foundin Khalid et al. (2004) and Edwards (2004), respectively.

This paper reports the numerical results obtained by theNational Research Council (NRC) of Canada — Institute forAerospace Research (IAR) for the supersonic projectilesexamined in the NATO RTO Applied Vehicle Technology(AVT) Panel Group 082 and the TTCP WPN TP2 KTa 2-19studies. The RTO/AVT study focused on a dual control-finmissile, while the TTCP study considered an elliptical multi-finned missile and a square cross-section single-finned missile.The missiles are described in the next section of the paper. Thisis followed by a description of the numerical procedure used atIAR and an analysis of the results. Finally, conclusions andrecommendations for future work are presented.

MISSILE CONFIGURATIONS

Elliptical MissileThe elliptical missile examined in the KTa 2-19 study,

shown in Figure 1, had a maximum cross sectional areaAelliptical = 8110 mm2, which is equivalent in area to a circle witha diameter of Delliptical = 101.6 mm. The maximum diameteralong the major axis of the cross-sectional ellipse was174.5 mm, and the maximum diameter along the minor axiswas 58.7 mm. The length of the missile was Lelliptical =7Delliptical. The missile had two sets of control surfaces: a frontset of wing-like structures and a rear set of fins. Both sets ofcontrol surfaces had sharp front and rear edges and roundedtops. Full details about the geometry of the elliptical missilecan be found in Allen (2004). The flow around this missileconfiguration was expected to contain shock–vortex, shock–shock, and shock–boundary layer interactions.

Five elliptical missile simulations were performed at afreestream Mach number of Ma = 2.5. These included caseswith angles of attack of α = 5° and 15°, and roll angles of φ= 0°and 45°, where φ = 0° corresponds to the configuration wherethe major axis is aligned with the horizontal and φ is positiveclockwise when viewed from the rear. In addition, onesimulation was performed for α = 20°, yaw angle β= 8.6°, and

φ = 0° with Ma = 2.5. The flow Reynolds number was Re =6560 / mm. The surface and flow pressures were examined atx/Lelliptical = 0.30, 0.60, 0.70, 0.85, and 0.95, as indicated inFigure 1, where x = 0 at the tip.

Dual Control-Fin MissileThe geometry the dual control-fin missile used in the

RTO/AVT study is shown in Figure 2. It consisted of two setsof four control fins, the first located immediately downstreamof the nose cone and the second located at the tail of the missilebody. The missile had a diameter Ddual of 66.0 mm and a lengthLdual = 15Ddual. The 3.8-mm-thick control fins were tapered attheir edges, as shown in Figure 2. Complete details can befound in Khalid et al. (2004).

Simulations were performed with the fins oriented in an “X”configuration at a 45° angle to the horizontal and vertical axes.The freestream Mach and Reynolds numbers were Ma = 1.75and Re = 6.56 × 104 / mm, which give a freestream temperatureand pressure of 288.2 K and 52.29 kPa. Two angles of attackwere modelled: α = 6° and α = 24°. At the higher angle ofattack, regions of separated flow with well-developed unsteadyvortex structures were expected, providing a difficult test casefor the steady-state CFD calculations. The velocity profileswere compared against experimental data at x/Ddual = 5.8 and13.8, as indicated in Figure 2, where x = 0 at the tip.

Square MissileThe geometry of the square cross-section missile examined

in the KTa 2-19 study is shown in Figure 3. It consisted of acircular nose cone attached to a square body. The width of thesides of the square body, Ssquare, was 94 mm, and the length ofthe missile was Lsquare = 13Ssquare. Configurations with the finslocated at the sharp corners (shown in Figure 3) and with thefins mid-way between the sharp corners (not shown) wereconsidered. Full details of the geometry can be found in Wilcox

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Canadian Aeronautics and Space Journal Journal aéronautique et spatial du Canada

Figure 1. Elliptical missile geometry.

Figure 2. Dual control-fin missile.

et al. (2004). The sharp corners of the square cross section wereexpected to induce strong vortex systems over the missile bodythat have the potential to impact upon the rear control-finsurfaces, depending on the angle of attack and the finorientation.

Six flight conditions examined in the KTa 2-19 study arereported in this paper. These include missile configurationswith corner-mounted and side-mounted fins, and roll angles φ=0°, 22.5°, and 45°, where φ = 0° corresponds to a diamond-shaped cross section and φ is positive clockwise when viewedfrom the rear. The freestream Mach number was Ma = 2.5, theangle of attack was α = 14°, the yaw angle was β = 0°, and theReynolds number of the flow was Re = 13 120 / mm. Thenumerical and experimental surface pressures were comparedat x/Ssquare = 5.5, 8.5, and 11.5, as indicated in Figure 3, wherex = 0 at the tip.

NUMERICAL PROCEDURE

The initial flow solutions obtained by IAR were producedusing the WIND CFD code, which evaluates second-order-accurate finite differences of the governing Navier–Stokespartial differential equations in conservative form (Bush et al.,1998). The explicit terms were calculated using Roe’s second-order flux-difference-splitting algorithm. Since the flow wassupersonic, the implicit terms were evaluated using aparabolized Navier–Stokes operator. The solution wasadvanced using a one-stage Runge–Kutta scheme. Theequation set was closed with the Spalart–Allmaras one-equation turbulence model, which was used to calculate theturbulent viscous terms. The maximum level of convergencewas typically obtained when the residuals of the Navier–Stokesequations were reduced by approximately two orders ofmagnitude.

The WIND code performed well for the simple flow fieldsaround the elliptical missile for all flow conditions and the dualcontrol-fin missile when α = 6°. But convergence problemswere encountered for the dual control-fin missile when α wasincreased to 24°, and for all the square-missile simulations.These cases corresponded to flows for which there were strongvortices located next to the missile surface. Attempts to resolvethese problems using different numerical schemes and a fullrather than a parabolized Navier–Stokes operator in the WINDcode were unsuccessful.

To overcome the convergence problems, the flightconditions for the dual control-fin and square missiles were

also modelled using the Structured Parallel Research Code(SPARC) developed by Magagnato (1999) at the University ofKarlsruhe. The SPARC computations used a second-orderaccurate finite-volume Navier–Stokes discretization methodbased on the Jameson–Schmidt–Turkel scheme and symmetricsuccessive over-relaxation LU-decomposition. The Spalart–Allmaras model was used to close the equation set, similar tothe WIND code. SPARC converged well for all cases, and theresults were in good agreement with experimental data.Therefore, only the SPARC results were reported for the dualcontrol-fin and square-missile data contained in this paper.

Structured multi-block meshes were used for all thesimulations. The number of grid points ranged from 3 millionfor the square missile computations to 4 million for theelliptical and dual control-fin missile computations. Thedistance of the first grid point from the surface of the missile y+

was less than 1.5 for the square and elliptical missiles and lessthan 5.0 for the dual control-fin missile. The base flows wereneglected, and all computations assumed steady-state flows.The grid convergence index for these flows was approximately0.04–0.05 for the square and elliptical missiles and 0.07 for thedual control-fin missile.

RESULTS AND DISCUSSION

Elliptical MissileThe pressure coefficients CP around the elliptical missile

body at x/Lelliptical = 0.30, 0.60, 0.70, 0.85, and 0.95 are shownin Figures 4–8. The cross sections at x/Lelliptical = 0.30 and 0.60are upstream of the control surfaces, while the cross sections atx/Lelliptical = 0.70 and 0.85 bisect the set of wing-like controlsurfaces and the cross section at x/Lelliptical = 0.95 bisects therear set of fins (see Figure 1). The position on the surface of themissile is given by θ, where θ = 0° corresponds to the top of themissile at the vertical axis, and is defined as positive clockwisewhen viewed from the rear, as indicated by the diagram in theupper left corner of each plot. The profiles shown in the figuresdo not include the fin surface pressures, leading to gaps in someof the curves. Wind tunnel data from Allen (2004) are alsoshown in the plots for all except the last case (Figure 8), forwhich experimental data were not available.

Figures 4 and 5 show the pressure coefficients for α = 5°and 15°, respectively, when Ma = 2.5, α = 5°, β = 0°, and φ= 0°.The simulations captured the sharp drop in pressure around theleft and right sides of the cross section, and the sharp change inpressure above and below the horizontal fins at x/Lelliptical = 0.70and 0.85. The numerical CP values were slightly less than thewind tunnel data on the upper and lower surfaces of the missile,especially at the upstream stations (x/Lelliptical = 0.30 and 0.60),but overall there was good agreement between the wind tunneland CFD results.

The agreement between the numerical predictions andexperimental results was also good at a roll angle of φ = 45°(Figures 6 and 7). In these simulations, the maximum pressurewas located at the bottom right corner of the elliptical cross

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Figure 3. Square missile geometry.

section at θ = 145°. Again, the numerical CP values wereslightly less than the experimental data on the lower surface ofthe missile at the upstream stations. Also, the simulations didnot reproduce the sharp pressure rise at θ ~ 135° for both Machnumbers when x/Lelliptical = 0.95; this corresponds to a locationin the wake immediately behind the right wing-like controlsurface. When a yaw angle β = 8.6° was introduced (Figure 8),the pressure underneath the missile changed from anapproximately uniform distribution to one with a peak pressurelocated just underneath the right side of the missile. Because ofthe high angle of attack (α = 20°), the difference between themaximum and minimum pressure coefficients was muchgreater than that shown in Figure 4 with α = 5°.

Contour plots of the vorticity at the same axial locations areshown for the α = 5°, β = 0°, φ= 0° and the α = 15°, β = 0°, φ=45° cases in Figure 9. These two plots are representative of the

five elliptical missile cases examined in this study. At the lowerangle of attack, high values of vorticity developed in the flowabove the missile, especially above the set of wing-like controlsurfaces at x/Lelliptical = 0.85. The wakes behind these surfaceswere clearly visible at x/Lelliptical = 0.95. At the higher angle ofattack with a roll angle of φ = 45°, vortices were visible abovethe elliptical section. As these vortices developed (x/Lelliptical =0.85), they drifted upwards slightly and pulled high-energyfluid from the outer flowfield towards the inner regions of theupper missile surface. Strong interference was evident betweenthese vortices and the rear set of fins.

The predicted axial force (CX), normal force (CY), andpitching moment (CM) coefficients are listed in Table 1. Thenormalizing area was the maximum cross-sectional area of theelliptical missile Aelliptical, the normalizing length was Delliptical(the diameter of a circle with an area equivalent to Aelliptical),

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Figure 4. Surface pressures on the elliptical missile for M = 2.5, α = 5°, β = 0°, and φ = 0°.


and the pitching moment was taken about x/Lelliptical = 0.60. Thesimulations at φ= 45° had a smaller normal force ared with thesimulations at φ= 0°, while the simulation at β = 8.6° producedthe highest normal force. However, the axial force changed byless than 7% between all of the simulations. Moderate pitchingmoments were created when φ = 45°. Comparisons with windtunnel results (Allen, 2004) are also shown in Table 1 for β= 0°and φ = 0°. The agreement between the numerical andexperimental axial and normal forces was within 7% for bothcases, and both the experimental and numerical pitchingmoments were small. Wind tunnel force and momentcoefficients were unavailable for the other flight conditions.

Dual Control-Fin MissileProfiles of the x- and y-components of the velocity, U and V,

above the dual control-fin missile are shown in Figure 10 forthe α = 24° case. Curves are shown for x/Ddual = 5.8 (black,located behind the front set of fins) and 13.8 (grey, located infront of the rear set of fins). The velocities have beennormalized by the freestream velocity, U∞. The profiles begin atthe missile surface, y/Ddual = 0.5, and extend radially outwardsfrom the missile. Wind tunnel results from Leopold et al.(2003) are also provided. The CFD predictions were inreasonable agreement with the experimental results, althoughthere were differences of up to 0.20U∞ in both velocitycomponents at the upstream station between 0.9 < y/Ddual = 1.5.There was better agreement between the CFD results and the

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wind tunnel data at the station located farther downstream,x/Ddual = 13.8.

The Mach number contours predicted along the verticalplane bisecting the missile centreline for α = 24° are shown inFigure 11. A shadowgraph obtained by Leopold et al. (2003)for the same flow conditions is also shown for comparison. Thebow shock wave and the strong shock waves extending belowthe fins were well resolved by the computations.

The numerical lift, drag, and pitching moment coefficientsare compared against the wind tunnel data, which werecollected over a wide range of α values, in Figure 12. The setsof data are also in good agreement.

Square MissileThe predicted surface pressure coefficients CP around the

square missile with corner fins at x/Ssquare = 5.5, 8.5, and 11.5for different roll angles are shown in Figures 13–15. Theplanes at x/S = 5.5 and 8.5 are upstream of the fins, while theplane at x/S = 11.5 bisects the fins (see Figure 3). The positionθ = 0° corresponds to the top of the missile at the vertical axis,and is defined as positive clockwise when viewed from the rear,as indicated by the diagrams shown above each figure. Thepressure profiles shown for x/S = 11.5, which intersects the fins,only include the surface pressures on the body, not on the fins.Wind tunnel data (Wilcox et al., 2004) are also shown in theplots for comparison purposes. The wind tunnel CP data were

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Figure 8. Surface pressures on the elliptical missile for M = 2.5, α = 20°, β = 8.6°, and φ = 0°.

Figure 9. Absolute vorticity contours on the elliptical missile at specified planes: (a) M = 2.5, α = 5°, β = 0°, and φ = 0°; (b) M = 2.5, α = 15°, β = 0°, and φ =45°. Contours are shown for every 200 s–1.

Flight conditions

Ma α (°) β (°) φ (°) CFD CX Expt CX % diff CFD CY Expt CY % diff CFD CM Expt CM2.5 5 0 0 0.198 0.188 5.0 1.08 1.15 6.5 0.023 –0.0502.5 15 0 0 0.192 0.182 5.2 3.94 4.10 4.1 0.005 0.1002.5 5 0 45 0.204 — — 0.656 — — 0.121 —2.5 15 0 45 0.201 — — 2.39 — — 0.374 —2.5 20 8.6 0 0.205 — — 5.46 — — –0.018

Table 1. Comparison of numerical and experimental aerodynamic coefficients for the elliptical missile.

only measured for a body alone (without fins) configuration.This should have little effect on the stations upstream of thefins since the flow was supersonic; however, the wind tunnelCP results cannot be used at the x/S = 11.5 plane, and were,therefore, omitted from the figures.

The SPARC predictions were in good agreement with thewind tunnel data for all three roll angles. Some discrepanciesoccurred near 45° < θ < 70° and 290° < θ < 315° at the x/S = 8.5station for all three roll angles, as well as near the bottomcorners of the x/S = 5.5 station when φ = 22.5° and 45°.However, for the most part, the simulations captured the changein pressure across the sharp corners of the square missile bodythat was measured in the wind tunnel. The predicted force andpitching moment coefficients of the missile are compared withthe wind tunnel results in Table 2. The average percentdifference between the numerical and experimental results forall six cases was less than 7.29%.

The evolution of the vortex systems that formed above thesharp corners of the square cross-section missile are illustratedin Figures 16–21. Each figure shows contours of vorticity atvarious x-plane slices in the flow field. In the followingdiscussion, “left” and “right” refer to the sides of the missilewhen viewed from the front.

For the missile at φ = 0° with corner-mounted fins(Figure 16), small vortices were visible attached to the missilesurface immediately above the sharp corners at x/S = 4.0. Thesedrifted upwards relative to the missile body due to the moderateangle of attack, and at x/S = 11.5, were attached to the surface atthe sharp corners only. The vortices remained well-separatedfrom the fin surfaces and thus should have little effect on thefin-control effectiveness. When a roll angle of φ = 22.5° wasintroduced (Figure 17), the right corner of the square crosssection was raised above the opposite left corner. The vortexthat formed above the raised right corner drifted upwards from

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Figure 10. Comparison of the numerical velocity predictions with wind tunnel data for the dual control-fin missile at α = 24°: (a) x-component, (b) y-component.

Figure 11. (a) CFD-predicted Mach number contours on the α = 24° dual control-fin missile at 0.05 intervals along the vertical plane bisecting the missilecentreline, (b) shadowgraph of the dual control-fin missile at α = 24° (from Leopold et al., 2003).

the missile body as before, but the vortex that formed above thelowered left corner (shown behind the missile in the figure)rolled up along the missile surface and remained attached to thebody for the full length of the missile. As a result, the upper-most fin sliced through a portion of this vortex. A smalladditional third vortex was also visible in this plot, whichformed above the top corner of the square cross section. At alarger roll angle of φ = 45° (Figure 18), two large vorticesformed above the top two corners and drifted above the missileas observed in the previous cases. There were also two smallervortices that formed along the sides of the missile above thelower two corners; these remained attached to the missilesurface. Even though the large upper two vortices were closerto the fins at x/S = 11.5 compared with the case with φ= 0°, theywere still convected past the control surfaces relativelyunscathed.

With side-mounted fins (see Figures 19–21), the vorticesformed as before, but there was considerable interferencebetween the vortices and the control surfaces. When φ= 0° and22.5°, the vortices that separated away from the missile body

were sliced in half by the fins. The vortices that remainedattached to the missile surface when φ = 22.5° and 45° alsocollided with the fins. Therefore, one can expect serious controlissues and unsteady loading on the fins for a missile with finsoriented in this fashion.

A complete aerodynamic analysis of this square-sectionmissile was performed by Birch and Cleminson (2004). Theyreported that the normal force of the square-section missile wasgreater than the normal force of an equivalent circular-sectionmissile, even accounting for the different platform areas. Thecomputed aerodynamic coefficients listed in Table 2 indicatethat the greatest normal force was obtained when the missilewas in a diamond-shaped configuration (φ = 0°), regardless ofthe location of the control fins. This also corresponded to theconfiguration with the greatest pitching moment. The axialforce remained relatively constant regardless of the missile orfin orientation.

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Figure 12. Comparison of computational and experimental aerodynamic coefficients for the dual control-fin missile.

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Figure 13. Comparison of surface pressure contours for the squaremissile with corner-mounted fins at φ = 0°.

Figure 14. Comparison of surface pressure contours for the squaremissile with corner-mounted fins at φ = 22.5°.

CONCLUSIONS

The evolving vortex systems surrounding elliptical, dualcontrol-fin, and square missiles were captured usingcomputational fluid dynamic simulations. Such results can beused to identify important flow characteriztics, such as vortexand control surface interactions that can have catastrophiceffects on the aerodynamic performance of a given missile. Inparticular, the simulations demonstrated that the evolvingvortex systems for the elliptical missile at a 45° roll angle and a15° angle of attack impacted with the rear set of fins. For thesquare missile, there was considerably less interferencebetween the vortex systems and the control surfaces when thefins were mounted on the corners rather than the sides of themissile cross section. For the dual control-fin missile, thevortices shed from the front set of control fins will not interferewith the rear control surfaces at the tested angles of attack.Overall, these results demonstrate the potential ofcomputational fluid dynamics for predicting complex flowfeatures at supersonic speeds.

ACKNOWLEDGEMENTS

Parts of this study were funded by a collaborative agreementbetween Defence R & D Canada - Valcartier and IAR. TheWIND simulations reported in this paper were performed aspart of the TTCP WPN TP2 KTa 2-19 and a NATO RTO PanelGroup 082 studies. The preliminary dual control-fin missilecomputations were performed by Angela Wong as part of herwork as a Masters student at IAR. The SPARC code was madeavailable by Franco Magagnato at the University of Karlsruhe.

REFERENCES

Allen, J. (2004). “Force, Surface, Pressure, and Flowfield Measurements onan Elliptical-Body Concept at Supersonic Speeds”. AIAA Pap. 2004–5450.

Birch, T.J., and Cleminson, J.R. (2004). “Aerodynamic Characteristics of aSquare Cross-Section Missile Configuration at Supersonic Speeds”. AIAAPap. 2004–5197.

Bush, R.H., Power G.D., and Towne, C.E. (1998). “WIND: The ProductionFlow Solver of the NPARC Alliance”. AIAA Pap. 98–0935.

Edwards, J. (2004). “Evaluation of Computational Methods withExperiment for Non-circular Missile Configurations: An Overview”. AIAAPap. 2004–5457.

Leopold, F., Demeautis, C., and Faderl, N. (2003). “ExperimentalInvestigations of the RTO Missile Configuration at High Angles of Attack”.French-German Research Institute of Saint Louis, Rep. PU 657/2003.

Khalid, M., Dujardin, A., Hennig, P., Leavitt, L., Leopold, F., Mendenhall,M., Prince, S., and Birch, T. (2004). “Application of Various TurbulenceModels to Investigate the Aerodynamic Performance of a NASA Dual ControlMissile”. AIAA Pap. 2004–5198.

Magagnato, F. (1999). “SPARC: Structured Parallel Research Code”.Department of Fluid Machinery, University of Karlsruhe, Karlsruhe, Germany.

Wilcox, F., Birch, T.J., and Allen, J. (2004) “Force, Surface, Pressure, andFlowfield Measurements on a Slender Missile Configuration with SquareCross-Section at Supersonic Speeds”. AIAA Pap. 2004–5451.

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Figure 15. Comparison of surface pressure contours for the squaremissile with corner-mounted fins at φ = 45°.

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φ = 0° φ = 22.5° φ = 45°

Cornerfins

Sidefins

Cornerfins

Sidefins

Cornerfins

Sidefins

Average %difference

CX CFD 0.528 0.527 0.521 0.522 0.523 0.522 7.29Experiment 0.480 0.501 0.475 0.500 0.476 0.499 7.29

CY CFD 4.090 3.824 3.702 3.461 3.226 2.968 4.12Experiment 4.125 4.056 3.808 3.770 3.196 3.166 4.12

CM CFD –27.34 –24.33 –24.45 –21.65 –20.56 –17.51 5.64Experiment –27.6 –26.2 –24.6 –24.2 –19.6 –19.4 5.64

Table 2. Comparison of numerical and experimental aerodynamic coefficients for the square missile.

Figure 17. CFD flow vorticity for the square missile with corner-mounted fins at φ = 22.5°. Contours shown at 500 s–1 intervals.

Figure 18. CFD flow vorticity for the square missile with corner-mounted fins at φ = 45°. Contours shown at 500 s–1 intervals.

Figure 20. CFD flow vorticity for the square missile with side-mountedfins at φ = 22.5°. Contours shown at 500 s–1 intervals.

Figure 21. CFD flow vorticity for the square missile with side-mountedfins at φ = 45°. Contours shown at 500 s–1 intervals.

Figure 19. CFD flow vorticity for the square missile with side-mountedfins at φ = 0°. Contours shown at 500 s–1 intervals.

Figure 16. CFD flow vorticity for the square missile with corner-mounted fins at φ = 0°. Contours shown at 500 s–1 intervals.

fluid dynamic analysis of non-traditional supersonic projectiles

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